Notes #5, ECE594I, Fall 2009, E.R. Brown 93 Free-Space Power Coupling for Two Special Cases: Radar and Radiometry Friis' Transmission Formulation Marconi was the pioneer for a new generation of electrical engineers working in the area of “wireless”. One of the truly brilliant amongst these was Friis working at Bell Laboratories in the 1920s and 30s. Among other things, Friis was the first to take advantage of the inherent nature of antennas as passive, reciprocal components, and treat the free-space propagation between a transmit antenna and receive antenna as a two-port “link”. This was done first and foremost for the wireless communications “link”, which we review here first to set the stage for the following RF and THz sensor (i.e., radar and radiometer) “link” formulation. The first step in Friis’ formulation is the concept of an effective aperture Aeff for the receiving antenna, prrinceffrecSAPεφθ⋅=),( (1) where Prec is the power available to the antenna for delivery to a load, ),(rrincSφθis the average Poynting vector for incoming radiation along the direction (θr.φr) in the spherical coordinates centered at the receiving antenna, and εp is the polarization coupling efficiency. Note that this expression applies only when ),(rrincSφθis aligned with the direction of the beam-pattern maximum. When there is misalignment, another factor is required which is the just the receive beam-pattern, prrincrrreffrecSFAPεφθφθ⋅⋅=),(),( . (2) Next, we suppose that this received Poynting vector is generated by a second, transmitting antenna. We can relate the received power to the properties of the transmitting antenna by: )(4),()(4),(),,(),(22rrFGPrrFDPrSSttttinttttradtttrrincτπφθτπφθφθφθ⋅≡⋅=≡ (3)Notes #5, ECE594I, Fall 2009, E.R. Brown 94 where the subscript "t" is for transmitting, Prad is the total radiated power, Pin is the power used to drive the transmitting antenna (in the matched case, equal to Prad), θt and φt are the spherical angles in the spherical coordinate system centered at the transmitting antenna, τ (r) is the power transmission function including all attenuation effects, and r is the distance (i.e., the “range” between transmitter and receiver. In writing (3) it is understood that Ft is taken in the direction (θt,φt) pointing towards the receiver, which is not necessarily the direction of the maximum of Ft. Substitution of (3) into (2) yields the relationship prrtttrtineffrecrrFFGPAPετπφθφθ⋅⋅= )(4),(),(2 (4) This can be simplified further in terms of the (ostensibly) known parameters of the receiving antenna using the relationships, 2effrrecrrrecout/A4GPDGPPλπ≡= (5) where Pout is the power delivered to the load of the receiving antenna. Substitution of (4) into (5) yields ptttrrrtrinoutrFFGGrPPετφθφθπλ⋅⋅⋅⎟⎟⎠⎞⎜⎜⎝⎛= )(),(),(421 (6) the expression commonly known as Friis' formula after its originator. It effectively treats the antenna combination like a two-port circuit with the pattern angular dependence and polarization dependence included explicitly. The term (λ/4πr)2 is called the free-space loss factor, which is of considerable practical and historical importance. Several theoreticians of the 19th century believed that radiation would decay faster than 1/r2 from a source. It was Hertz's observation of this 1/r2 dependence of radiation that encouraged the technology of "wireless."Notes #5, ECE594I, Fall 2009, E.R. Brown 95 Friis’ transmission for Radar For radar systems, the transmitter (in systems engineering often shortened to “Tx”) and the receiver (often shortened to “Rx”) have, in addition to free space, a body between them (i.e., the radar "target") that scatters electromagnetic radiation from the Tx to the Rx. To first order, some bodies (particularly round metallic ones) absorb practically none of the incident power and, instead, scatter it isotropically. Conceptually, we can then think of the body as a passive Rx/Tx combination that receives a power according to (1) above and transmits it isotropically, so that |),(S||),(S|APincinceffincφθσ≡φθ=rr, and (7) 224/|),(| rPSincscattπφθ=r, (8) where σ is the (target) scattering cross section and r2 is the distance between the scatterer and the observation point. We now assume that Sinc originates from a Tx antenna and Sscatt radiates back to a second (Rx) antenna to create an “echo” of received-aperture power Prec . In this case, )(4),(|),(|121rrFGPSttttinincτπφθφθ⋅=r and (9) pscattrrreffrec|),(S|),(FAP εφθφθ=r (10) where r1 is the distance between Tx and the scatterer, and εp is the fraction of the scattered power that has the same polarization characteristics as the Rx antenna. As in (5) above, we assume to know the Rx properties so that 2effrrecrrrecout/A4GPDGPPλπ≡= (11)Notes #5, ECE594I, Fall 2009, E.R. Brown 96 where Pout is the power delivered from the Rx antenna to its load. By substitution of (9) into (7), (7) into (8), (8) into (10), and (10) into (11), we find the relation ptttrrrtrinoutFFGGrrPrrPεφθφθπσλττ⋅⋅= ),(),()4()()(22213221 (12) This is the famous "bistatic" (two stationary point) radar transmission equation. In the special ("monostatic") case that the transmitter and receiver share a common antenna, r1 = r2, Gr = Gt, Fr = Ft, τ(r1) = τ(r2) = τ(r ), and (12) reduces to PinoutFGrrPrPεφθπλπστ⋅⋅⎟⎠⎞⎜⎝⎛⋅=22222)],([44)]([ (13) Like Friis' formula for communications, this treats the radar problem like a two-port equivalent circuit. But physically it differs from Friis' with the additional r-2 factor, leading to an overall r-4 dependence of Pout on Pin. This result is of great practical importance because it generally means that radar systems must transmit much higher power levels than communications systems to achieve a satisfactory received power for signal processing.Notes #5, ECE594I, Fall 2009, E.R. Brown 97 Example: One application for THz radar systems is short-range concealed object detection and imaging. This example calculates (a) the received power and (b) the background-limited signal-to-noise ratio (SNR) for a 600 GHz bistatic coherent radar located 1 m from the target. (a) To get the received power, we make the following practical assumptions (1) the transmit and receive feedhorns are located side-by-side in close